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**Polar Coordinates Practice Test**

**Directions:** A good score is eight correct.

1. In spherical coordinates, the position of a point is specified by

(a) two angles and a distance

(b) two distances and an angle

(c) three distances

(d) three angles

2. Suppose a point has the coordinates *(θ,r) = (π,3)* in the conventional (or mathematician’s) polar scheme. It is implied from this that the angle is

(a) negative

(b) expressed in radians

(c) greater than 360°

(d) ambiguous

3. Suppose a point has the coordinates *(θ,r)* = (π/4,6) in the mathematician’s polar scheme. What are the coordinates *(α,r)* of the point in the navigator’s polar scheme?

(a) They cannot be determined without more information

(b) (−45°,6)

(c) (45°,6)

(d) (135°,6)

4. Suppose we are given the simple relation *g(x) = x* . In Cartesian coordinates, this has the graph *y = x* . What is the equation that represents the graph of this relation in the mathematician’s polar coordinate system?

(a) *r = θ*

(b) *r* = 1/ *θ* , where *θ ≠* 0°

(c) *θ* = 45°, where *r* can range over the entire set of real numbers

(d) *θ* = 45°, where *r* can range over the set of non-negative real numbers

5. Suppose we set off on a bearing of 135° in the navigator’s polar coordinate system. We stay on a straight course. If the starting point is considered the origin, what is the graph of our path in Cartesian coordinates?

(a) *y = x* , where *x* ≥ 0

(b) *y* = 0, where *x* ≥ 0

(c) *x =* 0, where *y* ≥ 0

(d) *y = -x* , where *x* ≥ 0

6. The direction angle in the navigator’s polar coordinate system is measured

(a) in a clockwise sense

(b) in a counterclockwise sense

(c) in either sense

(d) only in radians

7. The graph of *r* = −3θ in the mathematician’s polar coordinate system looks like

(a) a circle

(b) a cardioid

(c) a spiral

(d) nothing; it is undefined

8. Suppose you see a balloon hovering in the sky over a calm ocean. You are told that it is at azimuth 30°, that it is 3500 meters above the ocean surface, and that the point directly underneath it is 5000 meters away from you. This information is an example of the position of the balloon expressed in a form of

(a) Cartesian coordinates

(b) cylindrical coordinates

(c) spherical coordinates

(d) celestial coordinates

9. Suppose we are given a point and told that its Cartesian coordinate is *(x,y)* = (0,−5). In the mathematician’s polar scheme, the coordinates of this point are

(a) (θ, *r* ) = (3π/2,5)

(b) (θ, *r* ) = (3π/2,−5)

(c) (θ, *r* ) = (−5,3π/2)

(d) ambiguous; we need more information to specify them

10. Suppose a radar unit shows a target that is 10 kilometers away in a southwesterly direction. It is moving directly away from us. When its distance has doubled to 20 kilometers, what has happened to the *x* and *y* coordinates of the target in Cartesian coordinates? Assume we are located at the origin.

(a) They have both increased by a factor equal to the square root of 2

(b) They have both doubled

(c) They have both quadrupled

(d) We need to specify the size of each unit in the Cartesian coordinate system in order to answer this question

**Answers:**

1. a

2. b

3. c

4. c

5. d

6. a

7. c

8. b

9. a

10. b

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